1. Introduction
It is well known that derivation is a function of an algebra which
generalizes certain features of the derivative operator. It gives an
interesting insight to understand the structure and local properties of
an algebra. The concept to study the structure of ring theory and
derivations though established long back, but got stimulated after
Posner in [1] described
some important results on the derivations of prime rings. A fundamental
problem in the theory of derivations is to determine all the derivations
on an algebra. Over the years, many important variants of derivation
have been presented. Among these variants, Lie derivation and its
generic extension is currently more interesting and attracting. Lie
centralizer is not so common but very meaningful variant. These variants
are being widely discussed now a days.
Let be a unital
algebra and the center of is denoted by . We denote the commutator
(Lie product) and Jordan product of by and
respectively for all . We say that
a ring is an -algebra ( is a field) if is an -vector space equipped with a bilinear
product. Let be an additive map. We say is a derivation (respectively Jordan
derivation) if
(respectively ), for all . For an element , the mapping
given by for all is called an inner derivation of induced by . Let be an
additive map. The map is said
to be right (left) centralizer if for all
. We say
that is an Jordan
centralizer if for all . An additive
map
is called a Lie centralizer if for
all .
The characterization of Lie centralizer of quaternion algebra is
given in [2]. The
characterization of Lie centralizers and their generalizations is now
widely studied on different kinds of algebras by many authors in ([3,4,5,6,7]). Fosner et
al. characterised Lie centralizers of triangular tings and nest algebras
in [3]. Ashrafi et al.
computed multiplicative generalized Lie -derivations of unital rings with
idempotents and -centralizers
generalized matrix algebras in [4] and [5] respectively. Fadaee et al. characterized
Lie centralizers at the zero product of generalized matrix algebras in
[6] and Lie triple
centralizers of generalized matrix algebras in [7]. In [8], Martindale described the standard form
of Lie derivation on certain primitive rings. Similar results have been
discussed on von Neumann algebras by Miers [9]. Mokhtari computed Lie derivations on
trivial extension algebras in [10]. It is obvious that every Lie derivation
is a generalized Lie derivation. Besides there are two different
definitions of generalized Lie derivations in literature. One is
introduced by Atsushi Nakajima [11] which is stated as: for an additive map
, we
say that is generalized Lie
derivation (also known as -Lie
derivation) if there exist a Lie derivation such that
for all
and the other is by Bojan Hvala [12] which is stated as: for an additive map , we say
that is generalized Lie
derivation if there exist a linear map such that
for all . Both definitions are equally being discussed by
mathematicians. Hvala discussed the generalized Lie derivations on rings
and proved that every generalized Lie derivation on a prime ring can be
written as the sum of a generalized derivation and a central map.
Benkovič [13] proved that
every generalized Lie derivation from a unital algebra onto a unitary
bimodule can be written in the sum of a generalized derivation and a
central map that vanishes on the commutators of the algebra. The
description of generalized Lie derivation of Lie ideals of prime
algebras and nonlinear generalized Lie derivation of some classical
triangular algebras are respectively given in [14, 15]. The generalized Lie derivations of prime
rings are discussed in [16] by using both definitions. Hvala’s
definition of generalized Lie derivation covers both generalized
derivations and -Lie derivations.
On the other hand, Nakajima’s definition is more favourable which
unifies the notions of Lie derivation and Lie centralizer. We will, in
particular focus on Nakajima’s definition to compute the matrix
representation of generalized Lie derivations of algebra of Octonion.
The discussion about local and -local derivation of Octonion algebra
have been discussed in [17]. Later, this discussion was extended to
Cayley algebras in [18].
Mainly, our focus in this article is to describe the matrix
representation as well as the characterization of Lie derivation of
Octonion algebras equipped with commutator product. Authors
characterized in [19], the
Lie triple derivations of algebra of tensor product of some algebra
and quaternion algebra.
Ghahramani in [20] proved results on the
characterization of generalized derivation and generalized Jordan
derivation of ring of quaternion and in [21] discussed the characterization of Lie
derivation and its natural generic extension of quaternion ring.
This article is arranged in the following order: Section contains some minor details of Octonion
algebras equipped with commutator product denoted by . In Sections and , matrix representation of Lie
derivation as well as decomposition of Lie derivation of octonion
algebra in terms of Lie derivation and Jordan derivation of and inner derivation of is presented. Section contains the characterization of Lie
centralizer of Octonion algebras. In Section , the matrix representation of
generalized Lie derivation is computed.
2. The Octonion Algebra
Let be an arbitrary -torsion free unital ring. The octonion
algebra (denoted by )
over is a class of
non-associative algebra. It is a unital nonassociative algebra of
dimension with the basis
and the product defined in the following table.
<span id="corank1" label="corank1"
The table can be summarized as follows: where is the Kronecker delta and
is a completely
antisymmetric tensor with value +1 when .
An octonion is of the form with
real coefficients . By using
the product defined in the table given above, we can have the following
relations; Using the above product on the basis as Lie
product and extend it by linearity, we can equip this product on .
3. Lie Derivation of
Octonion Algebra
In this section, we compute matrix representation of Lie derivation
of the octonion Algebra. Let be a Lie
derivation. admits a matrix
representation with respect to the basis, which is an matrix whose entries are
defined by
Each column of is an element of .
Theorem 1. The algebra of Lie derivations of
Octonions is generated by the following matrices:
Proof. Let be a
Lie derivation of . Then
we write for some arbitrary . Applying on the identity for . So, we get for .
Applying on identities , and , we get
Now by applying on
and , we get
Now by applying on and , we get Similarly by applying on all the remaining identities, we get
for For , For , For , By comparing the coefficients, we get
for , for with and for with . Specifically, , , , , , and
. 
Theorem 2. Let . Let be a Lie derivation. Then can be written as .
Proof. Applying
on gives Substituting the values of , which is computed in
matrix representation of in above
theorem yields Summarizing the above expression yields our
required result. 
4. Characterizing
Lie Derivation of Octonion Algebra
Our next task is to present characterization of Lie derivations of
the algebra of Octonion. In Theorem of [21], it is shown that if be a -torsion free ring and be quaternion ring, then every Lie
derivation of can be decomposed
in terms of Jordan derivation and Lie derivation of and an inner derivation of , for every element . Here, we have:
Theorem 3. Let be a Lie
derivation. Then there exist an element in , a Lie derivation and a Jordan derivation on such that for every element .
Proof. Since is
an additive map, we can write
for some . It can be easily
seen that . Next,
we will fine with
, for arbitrary . Set .
Applying on , we get By comparing the coefficients, we get which implies Now, applying on the identities , , , , , , and
putting where is an additive map which is
uniquely determined by , we get
Next, let be arbitrary and put .
Applying on and using (),
we obtain By comparing the coefficients, we get Applying
on the identities
where and taking for some additive map
uniquely determined
by , we get
and
Replacing
by in (), for some
, we infer that
is a Lie derivation of . Moreover, applying on the identity and using the foregoing calculations, we can see
that is a Jordan derivation.
Now let be an arbitrary element. Using (), ()
and (), we find that where It can be easily verified that where and Consequently, 
5. Lie Centralizer of
Octonion Algebra
This section contains the characterization of Lie centralizer of
octonion algebra. In Theorem of
[2], it is shown that if
is a -torsion free unital ring and is quaternion ring. Then every Lie
centralizer of can be represented
in terms of a Lie centralizer and Jordan centralizer of . Here, we have:
Theorem 4. Let be a Lie
centralizer. Then there exists a Lie centralizer and a Jordan centralizer on such that for every element .
Proof. We have already assumed the form for some . Since is a Lie centralizer, we have Furthermore, By comparing the coefficients, we have ,
,
,
,
,
,
which reduces and to Applying
on the identities , , , , , we get
Now assume that, .
We have ,
which implies .
Application of on the identity
gives , which implies . Let
be an arbitrary then .
From this we get, .
Let and set .
Applying on , we get ,
which reduces to . Now
applying on the identities
,
,
,
,
,
,
and taking , where
is an additive map
which is uniquely determined by , we get
Our next goal is to calculate for arbitrary . Set .
Applying on and and putting , where is an additive map which
is uniquely determined by , we
get
Since is
a Lie centralizer, () implies that is a Lie centralizer on . Let . Applying on the identity and using () and (), we get shows that
is a Jordan centralizer on . Now
let be
an arbitrary element in . By ()
and (), we get ,
which completes the proof. 
6. Generalized
Lie Derivation of Octonion Algebra
Generalized derivation is an extension of natural derivation. It has
many applications in the literature since it is quite helpful in the
geometric classification of rings and algebras. In this section, we
compute the matrix representation of generalized Lie derivation of the
octonion algebra. Let be a generalized Lie derivation. admits a matrix representation with
respect to the basis, which is an matrix whose entries are
defined by
Each column of is an element of .
Theorem 5. Let be a
generalized Lie derivation of and be the basis of . Then the matrix
representation of is as follows
Proof. Let be a
generalized Lie derivation of , then
where is
the derivation of the octonion algebra.
Put , then
for . So, by using the
equation(), we get for .
Unlike the procedure of finding the Lie derivation of , we don’t need to verify
equation () for the products between the octonionic
units to compute the matrix representation of . Hence generalized Lie derivation is
much easier to compute once a Lie derivation is obtained.
Suppose that Applying on the identity for and comparing the coefficients, we
get for
with .
By using the same technique proposed in the previous theorem, we
get, for For For For For For For By comparing the coefficients, we get
for and . 
Theorem 6. Let . Let be a generalized Lie derivation with respect to
then can be written as .
Proof. Applying
on gives Substituting the values of , which is computed in
matrix representation of in above
theorem yields Summarizing the above expression yields our
required result. 
Example 1. Let an arbitrary element . Let
and otherwise in Lie
derivation of then
will be
Select and in the generalized Lie
derivation, then
Put
in ,
we get The left hand side of the above equation will be
By using (), we get
and
using (), we get .
Then by direct calculation the right hand side will be .
7. Conclusion
Lie algebras of derivations and its variants explore the nature of
given algebras. In this research article, we have described of matrix
representation as well as the characterization of Lie derivation of
Octonion algebra. The characterization of Lie centralizer of Octonion
algebra is also presented. We have also computed the matrix
representation of generalized Lie derivations of Octonion algebras.
Acknowledgements
This research has been supported by HEC of Pakistan under the project
NRPU 13150. Authors are thankful to HEC of Pakistan for supporting and
funding this research.
Conflict of Interest
The authors declare no conflict of interests.